• PROJECT CODE: J1-70047
• PROJECT TITLE: Strongly Regular Graphs and CFSG-free Arguments in Algebraic Graph Theory
• PROJECT TEAM: Balazs David, PhD; Prof. Dragan Marušič, PhD
• PERIOD: 1. 3. 2026 – 28. 2. 2029
• BUDGET: 450,000,00 EUR
• FINANCING: Slovenian Research and Innovation Agency (ARIS)
• PROJECT COORDINATOR: University of Primorska, Andrej Marušič Institute (Slovenia)
• PARTNERS: InnoRenew CoE, UP IAM (Slovenia); Faculty of Education, University of Ljubljana (Slovenia)

As a meeting point of combinatorics, geometry and algebra, strongly regular graphs have been in the interest of mathematical community for quite a long time, with first papers dating several decades ago. From the algebraic combinatorics viewpoint, they can be seen as a combinatorial aproximation of rank 3 graphs, that is, orbital graphs of transitive permutation groups of rank 3 – groups with point stabilizers having two additional orbits beside the fixed point. Of course, while the latter have been completely classified as a consequence of the Classification of Finite Simple Groups (CFSG, hereafter), a classification of the whole class of strongly regular graphs is presently beyond our reach. Different directions can be taken when looking for restricted classes of strongly regular graphs. One approach is the concept of t-vertex condition where the numbers of certain subgraphs containing a given pair of vertices are required to be invariant of the graph. Another important restriction imposed on strongly regular graphs is via the concept of k-isoregularity, where it is required that any two subsets of cardinality at most k and inducing isomorphic subgraphs have the same number of neighbors. Both of these directions meet through the concept of (m,n)-regularity, where in particular graphs satisfying the t-vertex condition coincide with (2,t)-regular graphs and k-isoregular graphs coincide with (k,k+1)-regular graphs. An additional important motivation for this project stems from the connection of strongly regular graphs to transitive permutation groups of rank 3, which are all known thanks to CFSG, as mentioned above. But many working in permutation groups (and by extension in algebraic graph theory) are of the opinion that CFSG should be used somewhat more cautiously and conservatively and that whenever possible, one should look for a direct proof, one that is CFSG-free. To this end, together with various other group-theoretic and combinatorial tools, the two concepts mentioned above – t-vertex condition and kisoregularity – will play an essential role. The following are three main goals of this project: (1) To obtain structural results about strongly regular graphs (and more generally about association schemes), in particular about their respective subclasses of (2,4)-regular and (3,4)-regular graphs. (2) To prove (without the use of CFSG) that for n>8 even no non-trivial 3-isoregular (strongly regular) n-bicirculant exists. (3) To prove (without the use of CFSG) that for n odd a non-trivial strongly regular n-bicirculant X does not satisfy the 4-vertex condition unless n=5 and X is either the Petersen graph or its complement.